Diophantus

Diophantus of Alexandria (c. 201 - 285 AD) sometimes called "the father of algebra", was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica (c. 250 AD), many of which are now lost. Diophantus was the first Greek mathematician who recognized fractions as numbers, thus allowed positive rational numbers for the coefficients and solutions.

Perhaps the topic [of this book] will appear fairly difficult to you because it is not yet familiar knowledge and the understanding of beginners is easily confused by mistakes; but with your inspiration and my teaching it will be easy for you to master, because clear intelligence supported by good lessons is a fast route to knowledge.

Following the dedication to Dionysus as quoted by Paul Drijvers, Secondary Algebra Education (2011)

If we arrive at an equation containing on each side the same term but with different coefficients, we must take equals from equals until we get one term equal to another term. But, if there are on one or on both sides negative terms, the deficiencies must be added on both bides until all the terms on both sides are positive. Then we must take equals from equals until one term is left on each side.

As a square number is known to be the product of a number multiplied by itself, so every polygonal number, multiplied by one number and added to another, both of which depend upon the number of its angles, produces a square number. I shall prove this, and shall show also how from a given side to find its polygon and conversely. Some auxiliary propositions must first be proved.

The solution in integers, or in rational numbers, of indeterminate equations belongs to diophantine analysis. The name honors Diophantus, whose treatise of thirteen books, of which only six survive, was the first on the subject. The Latin translation (A.D. 1621) of this suggestive fragment directly inspired Fermat to his creation of the modern higher arithmetic. It also inspired something much less desirable. Diophantus contented himself with special solutions of his problems; the majority of his numerous successors have done likewise, until diophantine analysis today is choked by a jungle of trivialities bearing no resemblance to cultivated mathematics. It is long past time that the standards of Diaphantus be forgotten though he himself be remembered with becoming reverence.

This work of Diaphantus... was the first Greek mathematics, if indeed it was Greek, to show a genuine talent for algebra. ...He had begun to use symbols operationally. This long stride forward is all the more remarkable because his algebraic notation... was almost as awkward as Greek logistic. That he accomplished what he did with the available techniques places him beyond question among the great algebraists.

If his works were not written in Greek, no one would think for a moment that they were the product of Greek mind. There is nothing in his works that reminds us of the classic period of Greek mathematics. His were almost entirely new ideas on a new subject. In the circle of Greek mathematicians he stands alone in his specialty. Except for him, we should be constrained to say that among the Greeks algebra was almost an unknown science.

In this work [Arithmetica] is introduced the idea of an algebraic equation expressed in algebraic symbols. His treatment is purely analytical and completely divorced from geometrical methods.

Florian Cajori, A History of Mathematics (1893).

He appears to be the first who could perform such operations as (x−1)×(x−2){\displaystyle (x-1)\times (x-2)} without reference to geometry. Such identities as (a+b)2=a2+2ab+b2{\displaystyle (a+b)^{2}=a^{2}+2ab+b^{2}}, which with Euclid appear in the elevated rank of geometric theorems, are with Diophantus the simplest consequences of the algebraic laws of operation.

Florian Cajori, A History of Mathematics (1893).

In the solution of simultaneous equations Diophantus adroitly managed with only one symbol for the unknown quantities and arrived at answers, most commonly, by the method of tentative assumption, which consists in assigning to some of the unknown quantities preliminary values, that satisfy only one or two of the conditions. These values lead to expressions palpably wrong, but which generally suggest some stratagem by which values can be secured satisfying all the conditions of the problem.

The "porisms" of Pappus anticipated projective geometry, the problems of Diophantus prepared the ground for the modern theory of equations.

Tobias Dantzig, Number: The Language of Science (1930).

Diophantus was the first Greek mathematician who frankly recognized fractions as numbers. He was also the first to handle in a systematic way not only simple equations, but quadratics and equations of higher order. In spite of his ineffective symbolism, in spite of the inelegance of his methods, he must be regarded as the precursor of modern algebra. But Diophantus was the last flicker of a dying candle.

Fermat, commenting about 1637 on Diophantus II, 8 (to solve x2 + y2 = a2), stated that "it is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or in general any power higher than the second into two powers of like degree; I have discovered a truly remarkable proof which this margin is too small to contain." This theorem is known as Fermat's last theorem.

Diophantus himself, it is true, gives only the most special solutions of all the questions which he treats, and he is generally content with indicating numbers which furnish one single solution. But it must not be supposed that his method was restricted to these very special solutions. In his time the use of letters to denote undetermined numbers was not yet established, and consequently the more general solutions which we are now enabled to give by means of such notation could not be expected from him. Nevertheless, the actual methods which he uses for solving any of his problems are as general as those which are in use to-day; nay, we are obliged to admit, that there is hardly any method yet invented in this kind of analysis of which there are not sufficiently distinct traces to be discovered in Diophantus.

Leonard Euler, Opera Omnia, I, II, p. 429-430, as translated by Heath, Diophantus of Alexandria, a Study in the History of Greek Algebra: With a Supplement Containing an Account of Fermat's Theorems... (1910); quoted by W.T. Sedgwick, A Short History of Science (1918); cited by John Stillwell, Numbers and Geometry (2012)

There is scarcely any one who states purely arithmetical questions, scarcely any who understands them. Is this not because arithmetic has been treated up to this time geometrically rather than arithmetically? This certainly is indicated by many works ancient and modern. Diophantus himself also indicates this. But he has freed himself from geometry a little more than others have, in that he limits his analysis to rational numbers only; nevertheless the Zetcica of Vieta, in which the methods of Diophantus are extended to continuous magnitude and therefore to geometry, witness the insufficient separation of arithmetic from geometry. Now arithmetic has a special domain of its own, the theory of numbers. This was touched upon but only to a slight degree by Euclid in his Elements, and by those who followed him it has not been sufficiently extended, unless perchance it lies hid in those books of Diophantus which the ravages of time have destroyed. Arithmeticians have now to develop or restore it. To these, that I may lead the way, I propose this theorem to be proved or problem to be solved. If they succeed in discovering the proof or solution, they will acknowledge that questions of this kind are not inferior to the more celebrated ones from geometry either for depth or difficulty or method of proof: Given any number which is not a square, there also exists an infinite number of squares such that when multiplied into the given number and unity is added to the product, the result is a square.

Many scholars go right through the period of their mathematical education without being introduced to indeterminate (Diophantine) problems. This should seem somewhat strange to us when we reflect on how many real problems of life require solutions which are meaningless unless they are whole numbers. The absence of Diophantine analysis from school curricula is difficult to justify. The methods... are not beyond the capabilities of schoolchildren. It is true that the problems often involve tedious calculations, but this should not be used as an excuse for ignoring the valuable principles of this analysis. Diophantus himself was usually content with finding just one solution... But once one solution has been obtained, the general solution can readily be obtained from it.

Graham Flegg, Numbers: Their History and Meaning (1983)

In 130 indeterminate equations, which Diophantus treats, there are more than 50 different classes... It is therefore difficult for a modern, after studying 100 Diophantic equations, to solve the 101st; and if we have made the attempt, and after some vain endeavours read Diophantus' own solution, we shall be astonished to see how suddenly he leaves the broad high-road, dashes into a side-path and with a quick turn reaches the goal, often enough a goal with reaching which we should not be content; we expected to have to climb a toilsome path, but to be rewarded at the end by an extensive view; instead of which, our guide leads by narrow, strange, but smooth ways to a small eminence; he has finished! He lacks the calm and concentrated energy for a deep plunge into a single important problem; and in this way the reader also hurries with inward unrest from problem to problem, as in a game of riddles, without being able to enjoy the individual one. Diophantus dazzles more than he delights. He is in a wonderful measure shrewd, clever, quick-sighted, indefatigable, but does not penetrate thoroughly or deeply into the root of the matter. As his problems seem framed in obedience to no obvious scientific necessity, but often only for the sake of the solution, the solution itself also lacks completeness and deeper signification. He is a brilliant performer in the art of indeterminate analysis invented by him, but the science has nevertheless been indebted, at least directly, to this brilliant genius for few methods, because he was deficient in the speculative thought which sees in the True more than the Correct. That is the general impression which I have derived from a thorough and repeated study of Diophantus' arithmetic.

The geometrical algebra of the Greeks has been in evidence all through our history from Pythagoras downwards, and no more need be said of it here except that its arithmetical application was no new thing to Diophantus. It is probable, for example, that the solution of the quadratic equation, discovered first by geometry, was applied for the purpose of finding numerical values for the unknown as early as Euclid, if not earlier still. In Heron the numerical solution of equations is well established, so that Diophantus was not the first to treat equations algebraically. What he did was to take a step forward towards an algebraic notation.

The creation of the formal language of mathematics is identical with the foundation of modern algebra. ...As far as Greek sources are concerned, the special influence of the Arithmetic of Diophantus on the content, but even more so on the form, of this Arabic science is unmistakable. ...concurrently with the elaboration... of the theory of equations which the Arabs had passed on to the West, the original text of Diophantus began, as early as the fifteenth century, to become well known and influential. But it was not until the last quarter of the sixteenth century that Vieta undertook to modify Diophantus' technique in a really critical way. He thereby became the true founder of modern mathematics.

The Alexandrians used fractions as numbers in their own right, whereas the mathematicians of the classical period spoke only of a ratio of integers, not of parts of a whole and the ratios were used only in proportions. However, even in the classical period genuine fractions... as entities in their own right, were used in commerce. In the Alexandrian period, Archimedes, Heron, Diophantus, and others used fractions freely and performed operations with them. Though... they did not discuss the concept of fractions, apparently these were intuitively sufficiently clear...

Morris Kline, Mathematical Thought from Ancient to Modern Times (1972) p. 134

Another feature of Alexandrian algebra is the absence of any explicit deductive structure. The various types of numbers... were not defined. Nor was there any axiomatic basis on which a deductive structure could be erected. The work of Heron, Nichomachus, and Diophantus, and of Archimedes as far as his arithmetic is concerned, reads like the procedural texts of the Egyptians and Babylonians... The deductive, orderly proof of Euclid and Apollonius, and of Archimedes' geometry is gone. The problems are inductive in spirit, in that they show methods for concrete problems that presumably apply to general classes whose extent is not specified. In view of the fact that as a consequence of the work of the classical Greeks mathematical results were supposed to be derived deductively from an explicit axiomatic basis, the emergence of an independent arithmetic and algebra with no logical structure of its own raised what became one of the great problems of the history of mathematics. This approach to arithmetic and algebra is the clearest indication of the Egyptian and Babylonian influences... Though the Alexandrian Greek algebraists did not seem to be concerned about this deficiency... it did trouble deeply the European mathematicians.

Morris Kline, Mathematical Thought from Ancient to Modern Times (1972) p.144

Bachet de Méziriac remarked that any number (that is, positive integer) is either a square, or the sum of two, three, or four squares. He did not pretend to possess a proof. He found indications pointing to his statement in certain problems of Diophantus and verified it up to 325. In short, Bachet's statement was just a conjecture, found inductively. ...his main achievement was to put the question: How many squares are needed to represent all integers? Once this question is squarely put, there is not much difficulty in discovering the answer inductively.

No one has yet translated from the Greek into Latin the thirteen books of Diophantus, in which the very flower of the whole of arithmetic lies hid, the ars rei et census [the art of the evaluation of wealth or tax] which to-day they call by the Arabic name of Algebra.

For not more than six books [of Diophantus] are found, though in the proëmium he promises thirteen. If this book, a wonderful and difficult work, could be found entire, I should like to translate it into Latin, for the knowledge of Greek I have lately acquired would suffice for this.

The Greek mathematician Diophantus of Alexandria... it seems fairly probable... flourished about 250 A.D. ...We gather from the Arithmetica's introduction that it originally consisted of thirteen Books. Of these, only six have survived until now in Greek... The remaining seven were considered irretrievably lost until the recent discovery of four other, hitherto unknown books in Arabic translation, which, since it is attributed to Qustā ibn Lūqā, must have been made around or after the middle of the ninth century.

But how much more splendid the methods which reduce the problems which seem to be hardly capable of solution even with the help of surds in such a way that, while the surds, when bidden (so to speak) to plough the arithmetic soil, become true to their name and deaf to entreaty, they are not so much as mentioned in these most ingenious solutions.

Diophantos of Alexandria: A Study in the History of Greek Algebra (1885)[edit]

Diophantos' work is so unique among the Greek treatises which we possess, that he cannot be said to recall the style or subject-matter of any other author, except, indeed, in the fragment on Polygonal Numbers; and even there the reference to Hypsikles is the only indication we can lay hold of.

Diophantos lived in a period when the Greek mathematicians of great original power had been succeeded by a number of learned commentators, who confined their investigations within the limits already reached, without attempting to further the development of the science. To this general rule there are two most striking exceptions, in different branches of mathematics, Diophantos and Pappos. These two mathematicians, who would have been an ornament to any age, were destined by fate to live and labour at a time when their work could not check the decay of mathematical learning. There is scarcely a passage in any Greek writer where either of the two is so much as mentioned. The neglect of their works by their countrymen and contemporaries can be explained only by the fact that they were not appreciated or understood.

The reason why Diophantos was the earliest of the Greek mathematicians to be forgotten is also probably the reason why he was the last to be re-discovered after the Revival of Learning. The oblivion, in fact, into which his writings and methods fell is due to the circumstance that they were not understood. That being so, we are able to understand why there is so much obscurity concerning his personality and the time at which he lived. Indeed, when we consider how little he was understood, and in consequence how little esteemed, we can only congratulate ourselves that so much of his work has survived to the present day.

Between the time of Regiomontanus and that of Rafael Bombelli Diophantos was once more forgotten, or rather unknown. The Algebra of Bombelli appeared in 1572, and in the preface to this work the author tells us that he had recently discovered a Greek book on Algebra in the Vatican Library, written by a certain Diofantes... Though Bombelli did not carry out his plan of publishing Diophantos in a translation, he has nevertheless taken all the problems of Diophantos' first four Books and some of those of the fifth, and embodied them in his Algebra, interspersing them with his own problems. ...notation in this work of Bombelli's ...shows ...some advance upon that of Diophantos.

With the help of books only he [Wilhelm Xylander] studied the subject of Algebra, as far as was possible from what men like Cardan had written and by his own reflection, with such success that not only did he fall into what Herakleitos called... the conceit of "being somebody" in the field of Arithmetic and "Logistic," but others too who were themselves learned men thought him an arithmetician of exceptional merit. But when he first became acquainted with the problems of Diophantos his pride had a fall so sudden and so humiliating that he might reasonably doubt whether he ought previously to have bewailed, or laughed at himself. He considers it therefore worth while to confess publicly in how disgraceful a condition of ignorance he had previously been content to live, and to do something to make known the work of Diophantos, which had so opened his eyes.

It may be in some measure due to the defects of notation in his time that Diophantos will have in his solutions no numbers whatever except rational numbers, in [the non-numbers of] which, in addition to surds and imaginary quantities, he includes negative quantities. ...Such equations then as lead to surd, imaginary, or negative roots he regards as useless for his purpose: the solution is in these cases ὰδοπος, impossible. So we find him describing the equation 4 = 4x + 20 as ᾰτοπος because it would give x = -4. Diophantos makes it throughout his object to obtain solutions in rational numbers, and we find him frequently giving, as a preliminary, conditions which must be satisfied, which are the conditions of a result rational in Diophantos' sense. In the great majority of cases when Diophantos arrives in the course of a solution at an equation which would give an irrational result he retraces his steps and finds out how his equation has arisen, and how he may by altering the previous work substitute for it another which shall give a rational result. This gives rise, in general, to a subsidiary problem the solution of which ensures a rational result for the problem itself. Though, however, Diophantos has no notation for a surd, and does not admit surd results, it is scarcely true to say that he makes no use of quadratic equations which lead to such results. Thus, for example, in v. 33 he solves such an equation so far as to be able to see to what integers the solution would approximate most nearly.

The history of Alexandrian mathematics begins with the Elements of Euclid and closes with the Algebra of Diophantus, both of which are founded on the discoveries of several preceding centuries.

The solution of the higher indeterminates depends almost entirely on very favourable numerical conditions and his methods are defective. But the extraordinary ability of Diophantus appears rather in the other department of his art, namely the ingenuity with which he reduces every problem to an equation which he is competent to solve.

Diophantus shows great Adroitness in selecting the unkown, especially with a view to avoiding an adfected quadratic. ...The most common and characteristic of Diophantus' methods is his use of tentative assumptions which is applied in nearly every problem of the later books. It consists in assigning to the unknown a preliminary value which satisfies one or two only of the necessary conditions, in order that, from its failure to satisfy the remaining conditions, the operator may perceive what exactly is required for that purpose. ...a third characteristic of Diophantus [is] ...the use of the symbol for the unknown in different senses. ...The use of tentative assumptions leads again to another device which may be called... the method of limits. This may best be illustrated by a particular example. If Diophantus wishes to find a square lying between 10 and 11, he multiplies these numbers by successive squares till a square lies between the products. Thus between 40 and 44, 90 and 99 no square lies, but between 160 and 176 there lies the square 169. Hence x2=16916{\displaystyle x^{2}={\tfrac {169}{16}}} will lie between the proposed limits.

Sometimes... Diophantus solves a problem wholly or in part by synthesis. ...Although ...Diophantus does not treat his problems generally and is usually content with finding any particular numbers which happen to satisfy the conditions of his problems, ...he does occasionally attempt such general solutions as were possible to him. But these solutions are not often exhaustive because he had no symbol for a general coefficient.

Though the defects in Diophantus' proofs are in general due to the limitation of his symbolism, it is not so always. Very frequently indeed Diophantus introduces into a solution arbitrary conditions and determinations which are not in the problem. Of such "fudged" solutions, as a schoolboy would call them, two particular kinds are very frequent. Sometimes an unknown is assumed at a determinate value... Sometimes a new condition is introduced.

The Arithmetica... is deficient, sometimes pardonably, sometimes without excuse, in generalization. The book of Porismata, to which Diophantus sometimes refers, seems on the other hand to have been entirely devoted to the discussion of general properties of numbers. It is three times expressly quoted in the Arithmetica... Of all these propositions he says... 'we find it in the Porisms'; but he cites also a great many similar propositions without expressly referring to the Porisms. These latter citations fall into two classes, the first of which contains mere identities, such as the algebraical equivalents of the theorems in Euclid II. ...The other class contains general propositions concerning the resolution of numbers into the sum of two, three or four squares. ...It will be seen that all these propositions are of the general form which ought to have been but is not adopted in the Arithmetica. We are therefore led to the conclusion that the Porismata, like the pamphlet on Polygonal Numbers, was a synthetic and not an analytic treatise. It is open, however, to anyone to maintain the contrary, since no proof of any porism is now extant.

With Diophantus the history of Greek arithmetic comes to an end. No original work, that we know of, was done afterwards.

To give here an elaborate account of Pappus would be to create a false impression. His work is only the last convulsive effort of Greek geometry which was now nearly dead and was never effectually revived. It is not so with Ptolemy or Diophantus. The trigonometry of the former is the foundation of a new study which was handed on to other nations indeed but which has thenceforth a continuous history of progress. Diophantus also represents the outbreak of a movement which probably was not Greek in its origin, and which the Greek genius long resisted, but which was especially adapted to the tastes of the people who, after the extinction of Greek schools, received their heritage and kept their memory green.